Convergence analysis of GMRES for the Helmholtz equation via pseudospectrum
This work offers a theoretical framework for analyzing GMRES convergence for non-normal systems, which is important for preconditioner development in wave propagation problems.
The paper provides a priori convergence analysis of GMRES for non-normal linear systems arising from finite element discretizations of time-harmonic wave problems, deriving inclusion and exclusion regions for the pseudospectrum. The tools are applied to the Helmholtz equation with absorbing boundary conditions, with and without a shifted-Laplace preconditioner.
Most finite element methods for solving time-harmonic wave-propagation problems lead to a linear system with a non-normal coefficient matrix. The non-normality is due to boundary conditions and losses. One way to solve these systems is to use a preconditioned iterative method. Detailed mathematical analysis of the convergence properties of these methods is important for developing new and understanding old preconditioners. Due to non-normality, there is currently very little existing literature in this direction. In this paper, we study the convergence of GMRES for such systems by deriving inclusion and exclusion regions for the pseudospectrum of the coefficient matrix. All analysis is done a priori by relating the properties of the weak problem to the coefficient matrix. The inclusion is derived from the stability properties of the problem and the exclusion is established via field of values and boundedness of the weak form. The derived tools are applied to estimate the pseudospectrum of time-harmonic Helmholtz equation with first-order absorbing boundary conditions, with and without a shifted-Laplace preconditioner.